integral of (6x^5-sqrt(x)+5x^2)/(x^3)

解答

\int \frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} d x

2 x^{3} + \frac{2}{3 x ^{\frac{3}{2}}} + 5 ln ∣ x ∣ + C

求解步骤

\int \frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} d x

乘开

\frac{6 x ^{5} - x + 5 x ^{2}}{x ^{3}} : 6 x^{2} - \frac{1}{x ^{\frac{5}{2}}} + \frac{5}{x}

= \int 6 x^{2} - \frac{1}{x ^{\frac{5}{2}}} + \frac{5}{x} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 6 x^{2} d x - \int \frac{1}{x ^{\frac{5}{2}}} d x + \int \frac{5}{x} d x

\int 6 x^{2} d x = 2 x^{3}

\int \frac{1}{x ^{\frac{5}{2}}} d x = - \frac{2}{3 x ^{\frac{3}{2}}}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= 2 x^{3} - (- \frac{2}{3 x ^{\frac{3}{2}}}) + 5 ln ∣ x ∣

化简

= 2 x^{3} + \frac{2}{3 x ^{\frac{3}{2}}} + 5 ln ∣ x ∣

解答补常数

= 2 x^{3} + \frac{2}{3 x ^{\frac{3}{2}}} + 5 ln ∣ x ∣ + C

\int e^{- p x} sin (w x) d x

\int \frac{1}{2} sin (2) d x

\int \frac{- cos ( x )}{sin ^{2} ( x )} d x

\int (8 x + 3) e^{x} d x

\int 2 k x d x